Exact Green’s functions for delta-function potentials and renormalization in quantum mechanics

We present a simple recipe to construct the Green’s function associated with a Hamiltonian of the form H = H0 + λ δ(x), where H0 is a Hamiltonian for which the associated Green’s function is known. We apply this result to the case in which H0 is the Hamiltonian of a free particle in D dimensions. Field theoretic concepts such as regularization, renormalization, dimensional transmutation and triviality are introduced naturally in order to deal with an infinity which shows up in the formal expression of the Green’s function for D ≥ 2. Typeset using REVTEX ∗Present address: Instituto de F́ısica, Universidade de São Paulo, Cx. Postal 66318, CEP 05315970, São Paulo, SP. E-mail: [email protected]